Talk:Root of unity

Merge Root of unity modulo n

I think there's enough overlap between this article and Root of unity modulo n to warrant merging it with this article. Wqwt (talk) 01:30, 3 September 2018 (UTC)

• Oppose. Although related to Root of unity and Finite field#Roots of unity, this article is about a different case: the roots of unity in a ring that is not a field. I have added a link to Root of unity modulo n in the body of Root of unity. Nevertheless Root of unity modulo n would require to be cleaned up by adding context, explicit examples, and better explanation of the differences with the case of roots of unity in a field. D.Lazard (talk) 10:13, 2 October 2018 (UTC)
• better not! Jackzhp (talk) 13:19, 10 October 2018 (UTC)
• I agree that this seems like a bad idea, and will remove the tag. --JBL (talk) 23:52, 12 January 2019 (UTC)

Unclear statement

As the fourth bullet point in the section Explicit expressions in low degrees, this passage appears:

"*As Φ₅(x) = x⁴ + x³ + x² + x + 1, the four primitive fifth roots of unity are the roots of this quartic polynomial, which may be explicitly solved in terms of radicals, giving the roots

${\frac {\varepsilon {\sqrt {5}}-1}{4}}\pm i{\frac {\sqrt {10+2\varepsilon {\sqrt {5}}}}{4}},$

where $\varepsilon$ may take the two values 1 and –1."

But it needs to be stated (if true) that the two instances of ε are independent — or not, rather than left to the reader to wonder which is the case.2600:1700:E1C0:F340:38C4:1D3E:151:63CC (talk) 03:34, 2 October 2018 (UTC)

 Fixed (You could have done this edit yourself.) D.Lazard (talk) 08:37, 2 October 2018 (UTC)

Primitive root

IMHO nobody—except for Trente percente—cares about such technicalities. The term primitive root is explained in this article, not elsewhere. Hence it should be boldfaced one time. Incnis Mrsi (talk) 09:31, 9 September 2019 (UTC)

Explicit Expression of the 7th Root of unity

The real part of the 7th roots of unity can be expressed as the solutions of 8x³ + 4x² - 4x - 1 = 0, which can be overlaid with the unit circle to find all the roots. As is, the method to find the seventh roots is notably more complicated than this. — Preceding unsigned comment added by 96.240.140.67 (talk) 16:10, 2 March 2021 (UTC)

As a result of the symmetry of the seventh roots of unity, there are four distinct real parts. Your equation has only three solutions.—Anita5192 (talk) 17:30, 2 March 2021 (UTC)

Correction, I meant the primitive roots of unity. The missing fourth zero is x = 1. — Preceding unsigned comment added by 96.240.140.67 (talk) 17:48, 2 March 2021 (UTC)

The only difference between your method and that of the article is that you write the equation of the real part and the article writes the equation of twice the real part, that is the sum of a primitive root and its consjugate. The advantages of the method of the article are: the cubic equation is simpler and it makes clear that its roots are algebraic integers (this is often important in applications). So I see no reason for changing this section of the article. D.Lazard (talk) 18:22, 2 March 2021 (UTC)

Were the references listed twice?

@Anita5192: I noticed that all the links in the "notes" section were references in the article, so it seemed pointless to include them twice. I'm sorry for troubling you if I'm wrong, but it seems wrong to include duplicated information. — Preceding unsigned comment added by MEisSCAMMER (talk • contribs) 23:36, 15 March 2021 (UTC)

The Notes section contains citations of sources from the article. The References section contains mostly different sources. In this case, not all of the sources in the References section are cited, but that does not mean they should be removed. An editor may want to cite them in the future.—Anita5192 (talk) 00:11, 16 March 2021 (UTC)

Okay, my bad then. (Sorry for not signing that other comment -- I don't know what I was thinking.) MEisSCAMMER^(talk)_Hello! 12:15, 16 March 2021 (UTC)

Plot of complex functions

In § Trigonometric expression the plots of the complex functions $z\mapsto z^{3}-1$ and $z\mapsto z^{5}-1$ have been recently changed. It is unclear whether the new version is better. In any case, all these plots are difficult to interpret and of no help to understand the article, which is about equations $z^{n}-1=0,$ not functions $z\mapsto z^{5}-1.$ So MOS:PERTINENCE applies, and I'll remove these images. D.Lazard (talk) 09:00, 7 December 2021 (UTC)

Second root of unity

The article lists -1 as the only primitive second root of unity. To the best of my understanding, -i^2 is also 1, so shouldn't that be a second one?

Wanted to check rather than editing the article in case I'm misunderstanding something. 185.248.67.197 (talk) 19:19, 22 March 2022 (UTC)

$i^{2}=(-i)^{2}=-1.$ So, 1 and −1 are the only square roots of unity, and only −1 is a primitive second root of unity. i and −i are the two primitive fourth roots of unity. It seems that you confuse $-(i^{2})$ and $(-i)^{2}.$ D.Lazard (talk) 21:11, 22 March 2022 (UTC)

Thirds Roots of Unity Figure

In the figure "The 3rd roots of unity" W^1 and W^2 are reversed and need to be swapped. The figure here: https://commons.wikimedia.org/wiki/Category:Roots_of_unity#/media/File:3rd_roots_of_unity_correction.svg is correct. References: G0172 (talk) 16:12, 21 June 2022 (UTC)

There is nothing wrong in the figure, as W^1 and W^2 are both primitive roots of unity, and there is no reason (except common habits) for preferring the one with positive imaginary part. Nevertheless you could have changed the figure yourself. D.Lazard (talk) 17:34, 21 June 2022 (UTC)

References

Reducibility of polynomials of roots of unity

One section says the polynomials are “irreducible”, but I think it needs to be clarified.

If one writes the nth roots of unity in e notation, then the factors x – e^(-2kπi/n) and x – e^(2kπi/n) can be combined to give a polynomial over the real numbers. The sum of the two roots above is real because they are symmetrical about the real axis.

For example: (x^2 + φx + 1)(x^2 + φ^-1x + 1)(x - 1) = x^5 - 1 where φ = (1+ sqrt5)/2 (the golden ratio) Ziconium (talk) 17:04, 16 March 2023 (UTC)

This is the irreducibility over the integers (or equivalently, over the rational numbers) that is considered here. I have clarified this in the article. D.Lazard (talk) 17:47, 16 March 2023 (UTC)

Asking about my latest edit which got reverted

D.Lazard had reverted my latest edit. And whilst I don't mind the revert itself. I'd be glad if the revert reason can be elaborated further than simply WP:ELNO.

Best regards. Jothefiredragon (talk) 09:35, 8 January 2024 (UTC)

In WP:ELNO you will find several reasons for the revert. In WP:Reliable sources, you will find further reasons. On the other hand, you did not provide any reason for including this particular reference (their are thousands of similar courses, and I see no reason for emphasizing this particula one). D.Lazard (talk) 09:56, 8 January 2024 (UTC)